Hello Miss Molly, sounds like you are well on your way, nice one! Don't be surprised though if your dd seems to suck and chew everything with gusto this week but then the next is not interested, its quite normal and don't panic. Just continue offering bits of food when you wish and see what happens, its so much more fun than pureeing and spooning concoctions in to your little ones mouth!
In answer to your questions, I gave ds cooked egg from 6 months but he doesnt particularly like it! He's 15mths now and will sometimes eat egg mayo at nursery. I think the usual allergy rules apply, try it, keep an eye on it etc. re the food in nappy thing, this is totally normal, some foods will go straight through like raisins, leeks, sweetcorn, beans but its not a problem.
Re finger food, I found the easiest things at first to be brocolli, cooked and cooled potatoes, toast, pitta, softened shreddies/ mini weetabix at breakfast (softened with a splash of milk/ emb/ formula whatever you choose), cucmber slices. To be honest, anything that ds wouldnt choke on but could mess with as that's all he did for the first few months. Don't get too hung up on it, relax, give her something off your plate you think is suitable, share, its much easier this way. And remember "food is for fun until they are one!" so it doesnt matter if she is not eating much for the first months of weaning, its not important as long as she is getting her milk she will be fine.
Eventually you will see she starts to eat more and drink less, its not a forced or conscious issue by you, it just happens!
Sorry to witter on, its just that for the first few weeks of BLW'ing ds I was so obsessed with whether I was doing the right thing that I forgot the basics of BLW, about how stress free it should be and then as soon as I relaxed with it, it came much more naturally and easily!
Good luck!